Eur. Phys. J. E (2012) 35: 104
DOI 10.1140/epje/i2012-12104-0
THE EUROPEAN
PHYSICAL JOURNAL E
Regular Article
Diffusion of granular rods on a rough vibrated substrate
V. Yadava and A. Kudrollib
Department of Physics, Clark University, Worcester, MA 01610, USA
Received 5 June 2012 and Received in final form 4 September 2012
c EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2012
Published online: 17 October 2012 – Abstract. We investigate the effect of shape anisotropy and number density on the dynamics of granular
rods on a substrate with experiments using a mono-layer of bead chains in a vibrated container. Statistical
measures of the translational and rotational degrees of freedom indicate a dramatic slowing down of
dynamics because of steric interactions at a value well below the highest packing fraction of the chains.
In particular, the in-plane orientation auto-correlation function decays exponentially with time at low
densities, but increasingly slowly as density is increased with a form which is not described by a simple
exponential function. While the mean square displacement of the chain center of mass is observed to grow
linearly at low densities, it is observed to grow increasingly slowly and non-linearly as number density is
increased. Decomposing the displacements parallel and perpendicular to the long axis of the chain, we find
that the ratio of diffusion in the parallel and perpendicular direction to their long axis is less than one in
the dilute regime. However, as the density of the chains is increased, the ratio rapidly increases above one
with a greater value for higher aspect ratios. This anisotropic behavior can be explained by considering a
higher effective drag on the rods by the substrate in the perpendicular direction compared with the parallel
direction, and by tube-like dynamics at higher densities.
1 Introduction
A large variety of rod-like particles exist in nature ranging from molecular polymers to bacteria and cereal grains
which play a vital role in natural and industrial processes.
At the smallest scales, the particles are thermally excited
and their diffusion properties at equilibrium can be calculated using statistical mechanics [1]. However, it remains
unclear if such an approach can describe observations
in out-of-equilibrium systems such as granular materials
which require constant energy input to remain active.
In fact, significant far-from-equilibrium features have
been observed. For example, experiments with collection
of vibrated granular rods have shown formation of dynamic vortex structures [2], and giant number fluctuations
which have been described by active nematic model [3].
Details of how energy is injected can be significant, and
frictional interaction with the vibrated substrate can provide preferential kicks around the axis of the rod from
the point of contact towards the center of mass of the
rod [4]. This effect can lead to a ratchet-like convective
motion [5], and can be also exploited to fabricate selfpropelled particles and study their emergent and diffusion
properties [6,7].
a
b
e-mail: [email protected]
e-mail: [email protected]
Nonetheless, a number of recent studies with uniformly
vibrated quasi-2D granular system find properties consistent with those for corresponding thermally agitated particle system. For example, experimental investigation of
caging dynamics with spherical beads on a rough substrate found a number of robust features typically associated with dense molecular liquids and colloids including development of a plateau in the mean square displacement, and a Vogel-Fulcher relaxation [8]. By using
a linked bead chain in a vibrated container with a rough
substrate [9], preferential excitation along the axis as in
the case of rigid rods appears to be reduced. This occurs
because when a chain hits the vibrating plate at an angle relative to horizontal, it dissipates energy by bending
and by inelastic collisions among beads in the chain reducing the kicks along the long axis. The spatial structure
of such a single vibrated granular chain has been shown
to be well described over a large range of lengths by a persistent self-avoided random walk model used to describe
polymers [10]. Further, the diffusion of the center of mass
was observed to decrease inversely with the length of the
chain and consistent with Rouse dynamics.
Our goal in this paper is to discuss the diffusion of
athermal rod-like particles as a function of their aspect
ratio and density using chains on a vibrated substrate to
minimize the preferential excitation along the axis as in
rigid rods. Because the chains we use are shorter than their
Page 2 of 7
Eur. Phys. J. E (2012) 35: 104
persistence length they serve as a sufficiently good approximation for rod-like particles without any subsidiary phenomenon that arise due to interaction with substrate as
with rigid granular rods. We observe that the dynamics of
the overall system is determined substantially by the interaction of the particles with the substrate at low concentrations, whereas the topological constraint of non-crossing
of chains controls the dynamics at higher concentrations.
n
2
3
5
(a)
2 Experimental system
The chains used in the experiments are composed of beads
with diameter d = 3.2 mm which are linked with a loose
link whose length can vary from 0 to 1.1 mm. We use
chains with bead number n = 2, 3 and 5 to vary the
length L of the rod. The aspect ratio Ar = L/d of the
chains are accordingly found to be on average 2.34, 3.78
and 6.41. Because each link can bend so that neighboring bonds between the beads in the chain can be between
π/4 and −π/4, we choose a small number of beads to
ensure that the length of the chain is smaller than its persistence length of ≈ 10d [10]. To test our assumption on
approximation of granular chain as rods we calculated the
probability distribution of fluctuation of beads about the
line of best fit. We observed that for longest chain (i.e.
5 beads) in dilute regime the fluctuation was less than
1 bead diameter in 87% of the observations. In concentrated regime, 91% of the cases had fluctuation that was
less than 1 bead diameter. Smaller chains will have even
smaller fluctuation due to reduced number of links, and
therefore we assume that all the chain lengths used in this
study can be approximated as granular rods. The experimental setup consists of a circular container with diameter dbase = 28.5 cm (89d) mounted on an electromagnetic
shaker. A layer of 1 mm steel beads were glued to the surface of plate to make it rough. This ensures random kicks
with components in the lateral direction when the chain
collides with the substrate and results in greater in-plane
dynamics. The system is then vibrated sinusoidally at a
peak acceleration of 3g, where g = 9.8 m s−2 . The acceleration provides sufficient driving force not only in the
vertical direction to move the chain over the roughness
scale, but does not allow them to hop over each other.
While a rod can have three translational and three rotational degrees of freedom, one degree of freedom each is
suppressed as the rods do not travel significantly off the
substrate, and do not flip in a vertical plane.
We vary the number of chains in the system to change
the area fraction φ which is defined as
φ = nmd2 /d2base ,
Table 1. The area fractions φ∗ and φc corresponding to the
transition from the dilute to the semi-dilute and to the concentrated regimes.
(1)
where n is the number of beads in each chain and m is
the number of chains in the system. Accordingly, the area
fraction φ∗ corresponding to the transition from the dilute to the semi-dilute density is given by n/A2r , and the
concentrated regime corresponds to above φc = πn/4Ar .
The corresponding values for the chains used in our experiments are shown in table 1. While the chains used
φ∗
0.36
0.21
0.12
y
φc
0.67
0.62
0.61
(b)
R(t+1)
R(t)
x
Fig. 1. (a) An image of a chain with 5 beads in the circular
container with a rough horizontal substrate. (b) A schematic
diagram denoting the position vector of the center of mass
R(t) and R(t + 1) at time t and t + 1, respectively, along with
the corresponding orientation vector u(t) and u(t + 1). Solid
black and gray frames depict lab and body frame of reference,
respectively. The components of the displacements in the lab
and body frame are denoted by dashed lines of same color as
the frame.
Fig. 2. Sample trajectory of a chain with 5 beads corresponding to (a) dilute regime φ = 0.006 over time interval
Δt = 1200 s, (b) semi-dilute regime φ = 0.42 over Δt = 1200 s,
and (c) concentrated regime φ = 0.67 over Δt = 3600 s. The
diffusion of the chain slows down as φ increases, and the observation period was changed accordingly.
are identical and colored black, a single chrome plated
but otherwise similar chain is used as a tracer to measure position over long periods of time using a mega-pixel
digital camera. A typical image is shown in fig. 1 along
with a schematic representing the laboratory reference
frame and the body reference frame of the rod. Because
smaller chains have faster dynamics compared to larger
chains [10], chains with 2 beads were tracked at 20 frames
per second (fps) and the chains with 3 and 5 beads were
tracked at 10 fps. The images were then used to calculate
position and orientation of rods in the laboratory reference frame in the horizontal plane. Typical reconstructed
trajectories corresponding to various φ are shown in fig. 2.
Eur. Phys. J. E (2012) 35: 104
Page 3 of 7
1
The rods diffuse over the entire container at low density
but appear confined at the highest density. Because of the
tracer technique used here we confine our discussion to
dynamics. Spontaneous pattern formation has been discussed using rigid granular rods by others [11].
9
(2)
where ν is the concentration of rod-like particles in three
dimensions, and β is a constant. However, when the measured Dr is scaled with Dr0 and plotted against reduced
concentration in 2D (i.e. νL2 ), a collapse of reduced rotational diffusion coefficients for chains of different lengths
0.00024
0.06
0.12
0.18
0.24
0.30
0.36
0.42
0.54
0.60
7
6
CD(t)
Dr = βDr0 (νL3 )−2 ,
φ
8
*
5
3 Rotational diffusion
4
4/π2
0.1
1
9
1
(b)
10
100
φ
0.00036
0.06
0.12
0.18
0.24
0.30
0.42
0.48
0.54
0.60
8
7
CD(t)
6
5
4
*
4/π2
(c)
1
1
10
100
9
8
φ
7
CD(t)
We first examine the dynamics using the orientation of
the chain in the horizontal plane. We obtain the orientation of the chain by fitting the bead positions with
a line which minimizes its radius of gyration. If u(t) is
the unit orientation vector of the chain at time t, then
we can evaluate the orientation auto-correlation function
CD (t) = u(t) · u(t + t0 ). Figure 3(a-c) shows the decay
of CD (t) for chains with n = 2, 3, and 5 at various area
fraction. A star denotes the area fraction φ∗ , which corresponds to transition from dilute to semi-dilute regime.
The upper range of φ investigated in our experiments is
determined by the value where the chains are no longer
confined in a single layer but start to form double layers,
and only the φ = 0.66 for the 5 bead chain falls in the concentrated regime. As expected, we observe that the decay
is rapid at low densities but becomes slower as the density is increased due to interactions with other particles.
Except for the data in the concentrated regime, CD (t) is
observed to approach 4/π 2 , which corresponds to uncorrelated orientations in case of rods.
In fig. 4, we plot CD (t) after subtracting 4/π 2 versus time to examine the rapid initial decay. In the dilute
regime, we find that CD (t) decays as an exponential from
which a single relaxation time scale can be extracted. However, this trend is not observed in the semi-dilute regime
where chains interact more frequently and perhaps multiple time scales may be in play. This lack of single relaxation time scale becomes an issue when we try to define a rotational diffusion coefficient from CD (t) at high
concentrations. Therefore, we define a rotational diffusion
coefficient Dr to be inverse of a characteristic decay time
in which CD (t) decays to half of its initial value. In fig. 5,
we plot Dr as a function of φ for various Ar . For the same
φ, we find that Dr of a chain with larger aspect ratio is
smaller than that of a chain with smaller aspect ratio. This
is attributed to the fact that the rods with larger Ar will
have a larger probability of colliding and hindering the
rotational motion of nearby rods at fixed φ, thus reducing
the overall Dr .
Using a tube model, Doi and Edwards calculated [1]
that the rotational diffusion coefficient in the semi-dilute
regime, Dr , and dilute regime, Dr0 , in 3D are related by
(a)
6
0.0006
0.06
0.12
0.18
0.24
0.30
0.36
0.42
0.48
0.54
0.61
0.67
*
5
4
4/π2
1
10
t (sec)
100
1000
Fig. 3. The decay of the orientation auto-correlation function
for chains with (a) n = 2, (b) n = 3, and (c) n = 5. The star
indicates the area fraction where system goes from dilute to
concentrated regime, with φ∗ = 0.37, 0.21 and 0.12 for rods
with n = 2, 3 and 5, respectively.
(which is linearly proportional to the rod mass) is not observed (see fig. 6). This can be explained if we take into
account the rotation of rods along their long axis. In 3D, a
rod of length L disengages from the constraints of a tube
of radius a by moving a distance of length L/2 along its
long axis. Over the same time it rotates by an angle of
order a/L. In 2D, when a rod meets a constraint posed by
a tube it rotates about the point of contact due to the ro-
Page 4 of 7
Eur. Phys. J. E (2012) 35: 104
(a)
4
2
10
Dr (sec-1 )
0.1
CD (t)
4
2
*
0.01
1
n=2
n=3
n=5
4
0.1
2
0.001
0.0
0.5
1.0
1.5
t(sec)
2.0
6
0.001
(b)
4
CD (t)
φ
0.1
1
Fig. 5. Decay of the rotational diffusion coefficient Dr as
a function of area fraction for chains of different lengths.
The dashed vertical line represents φ corresponding to the
densest
√ possible packing of the bead chain in our system, i.e.
π/2 3 ≈ 0.91.
2
0.1
8
6
4
0.01
1
*
6
4
2
(a)
2
0
1
2
t(sec)
3
4
Dr /Dr0
0.01
5
0.1
n= 2
n= 3
n= 5
6
4
6
2
4
0.01
CD (t)
2
6
4
0.1
8
6
*
4
1
(c)
1
6
4
2
10 2
1/νL
100
1000
(b)
2
0
1
2 t(sec) 3
4
Dr /Dr0
0.01
5
Fig. 4. Semi-log plot of CD (t) for chains with (a) n = 2,
(b) n = 3 beads, (c) n = 5. In the dilute regime, the decay can
be described by an exponential fit, this is not in the semi-dilute
regime where the decay curve may have multiple time scales.
tational inertia of rod along its long axis hence decreasing
the overall angle traveled by the rod. Because the angle by
which a rod rotates is presumably inversely proportional
to its moment of inertia and hence to its mass, eq. (2) can
be modified as
Dr = βDr0 (1/νL2 − K/n)α .
(3)
We observe that for K = 0.6, all the normalized diffusion
curves collapse on to each other with an exponent of α =
2.0 in semi-dilute regime. Fitting the collapsed data with
a single parameter function (eq. (3)) in β, we find β ≈
13.68 which is greater than what is assumed by Doi and
Edwards who consider it to be of order unity. However,
it is noteworthy that the observed α is smaller compared
0.1
6
4
n= 2
n= 3
n= 5
2
0.01
Eqn. 3
6
4
0.1
1
10
2
1/νL - K/n
100
1000
Fig. 6. (a) The normalized rotational diffusion coefficient as
a function of concentration. Curves for chains of different Ar
do not converge on a single line as observed in case of thermal
rods. (b) The normalized rotational diffusion coefficient plotted against modified concentration. The curves for different
athermal chains fall on the same master curve with the same
exponent as predicted for thermal rods.
to values reported in 3D [1,12]. We note that we have
changed our control parameter from volume fraction, φ to
reduced concentration, i.e. νL2 to avoid introducing an
extra dependence on the number of beads in eq. (2).
Eur. Phys. J. E (2012) 35: 104
(a)
6
4
2
slope 1
1
MSD/l
2
φ
*
0.00024
0.06
0.12
0.18
0.24
0.30
0.36
0.42
0.48
0.54
0.1
0.01
0.1
10
(b)
t (sec)
10
100
slope 1
1
MSD/l2
1
φ
*
0.00036
0.06
0.12
0.18
0.24
0.30
0.36
0.42
0.48
0.54
0.60
0.1
0.01
10
MSD/l
2
1
(c)
1 t (sec) 10
slope 1
100
φ
*
0.0006
0.06
0.12
0.18
0.24
0.30
0.36
0.42
0.48
0.54
0.60
0.66
0.1
0.01
0.001
1
t (sec) 10
100
Fig. 7. The mean square displacement (MSD) as a function of
time for chains with (a) n = 2, (b) n = 3, and (c) n = 5. The
star indicates the area fraction where system goes from dilute
to concentrated regime (for values refer to fig. 3).
4 Translational diffusion
4.1 Translational diffusion in lab frame
We next turn to the analysis of the mean square displacement (MSD) of the rods in the laboratory frame of reference. If R(t ) is the position of center of mass of tracer
chain at time t , then MSD over a time t is defined by
ΔR2 (t) = (R(t + t0 ) − R(t0 ))2 . In fig. 7, we plot
ΔR2 (t) against time for various area fractions for chains
of different length. Here, the vertical axis is scaled over a
D(sec-1 )
10
Page 5 of 7
0.1
6
4
2
0.01
6
4
n=2
n=3
n=5
0.001
0.01
φ
0.1
1
Fig. 8. Decay of the total diffusion coefficient as a function of
area fraction for chains of different lengths. The dashed black
line represents φ corresponding to the densest possible packing
in our system, i.e. 0.91.
distance l = Rbase /5 over which a chain travels in the dilute regime before becoming diffusive. We observe at short
time scales that the motion is super-diffusive, which is a
signature of transition from ballistic to diffusive regime.
(By performing a limited set of experiments with a faster
than 1000 frame rate camera, we were able to verify that
a ballistic regime indeed exists over time scales shorter
than 20 ms.) At intermediate time scales, the motion of
chains is diffusive in the dilute regime but becomes more
and more sub-diffusive as the area fraction increases. The
MSD plots also shows two distinct time scales at higher
concentrations. Here, the slow mode is due to caging of
a chain by its neighbors, and the fast decay mode is due
to rapid motion after cage breaking. At long time scales,
we observe that MSD starts saturating at a length scale
corresponding to the size of the container.
As with rotational diffusion, it is difficult to determine
the translational diffusion constant from a linear fit of
ΔR2 (t). Therefore, we deduce a translational diffusion
coefficient D by defining it as the inverse of a characteristic time scale over which a particle diffuses by a distance
of l. The translational diffusion coefficient versus φ for
chains of different lengths is plotted in fig. 8. We observe
a systematic decrease in D with an increase in the number of beads in the chain at low densities. In a previous
work by Safford et al. [10] it has been shown that the
chains show Rouse dynamics, i.e. D decays as 1/n in the
dilute regime which is consistent with the observations
shown here. Further it can be noted that the rapid decrease of diffusion occurs in the semi-dilute regime, and
the chains appear caged well before the highest packing
fraction 0.91 possible for the chain system. In the case of
spheres, Reis et al. [8] observed that the particles appear
caged above φ ∼ 0.72 with a similar vibration system and
close to the crystallization transition. In our experiments
with non-spherical particles, caging appears to occur at a
lower value, and is at least as low as 0.66 in the case of
n = 5.
Page 6 of 7
τ|| (sec)
100
3.5
(a)
4
2
0.5
0.001
100
8
6
τ⊥ (sec)
4
0.01
0.1
0.0
1
3.0
(b)
0.1
0.2
(b)
φ
0.3
0.4
0.5
2.5
n=2
n=3
n=5
2
10
2.0
1.0
8
6
2
n= 2
n= 3
n= 5
2.5
1.5
D|| /D⊥
10
(a)
3.0
n=2
n=3
n=5
8
6
D|| /D⊥
2
Eur. Phys. J. E (2012) 35: 104
n=2
n=3
n=5
2.0
1.5
1.0
8
6
0.5
4
0.001
0.01
φ
0.1
1
Fig. 9. Variation of mean time required to diffuse through a
distance l in a direction (a) parallel and (b) perpendicular to
rod, respectively, as a function of area fraction. The dashed
black line represents φ corresponding to the densest possible
packing in our system, i.e. 0.91.
4.2 Translational diffusion in body frame
Any anisotropy of diffusion is not directly evident in laboratory frame due to the rotation of the rod in the horizontal plane. Therefore, we decompose the displacement
of the chain in a body frame attached to the center of
mass of the chain with an axis oriented along the long
axis of the chain which minimizes its radius of gyration
and the other axis oriented perpendicular to it. Again, we
can define a diffusion constant in a given direction to be
the inverse of a characteristic time that the particle takes
to diffuse by a distance of l in that direction. In fig. 9,
we plot the variation of characteristic time scales in parallel (τ ) and perpendicular (τ⊥ ) directions for chains of
various aspect ratios as a function of their area fraction.
Overall, we find similar trends for τ and τ⊥ with higher
time scales for higher aspect ratios. This can be attributed
to more frequent collisions among beads in chain along its
length. Further it may be noted that the time scales for
parallel and perpendicular components diverge together
and over apparently similar φ as the net diffusion.
To understand the differences, we obtain the ratio
D /D⊥ = τ⊥ /τ in the dilute and semi-dilute regime and
Dilute
0.001
0.01
Semi-Dilute
0.1
φ / φ*
1
Fig. 10. (a) Ratio of diffusion coefficient in parallel and perpendicular direction (D /D⊥ ) is plotted against area fraction
for rods of different lengths. (b) D /D⊥ is plotted against
scaled area fraction. It is clearly observed that D is less than
D⊥ in the dilute regime.
plot it in fig. 10(a) as a function of φ, and as a function
of φ scaled with corresponding φ∗ in fig. 10(b). We find
that the data organizes such that the D /D⊥ in the dilute regime is less than one for all chains, and increases
above one in the semi-dilute regime. In case of a randomly
excited rod moving in 2D with slip boundary conditions,
the ratio should be equal to one if the dissipation is independent of orientation with respect to its velocity. However, if the dissipation depends on orientation then the
ratio can be non-zero depending on the aspect ratio even
in case of thermal excitation [13]. Because we have used
semi-flexible chains in our experiments which do not get
preferential kicks along their axis as in case of rigid rods on
a frictional substrate [4], the relatively greater diffusion in
the perpendicular direction must be a result of lower dissipation compared to that in the parallel direction at low
densities. We believe this lower dissipation occurs because
of rolling of rods, which increases the diffusion in perpendicular direction. Close to φ∗ this rolling is arrested and
we observe that the ratio D /D⊥ tends to one for rods
of all aspect ratios. In the semi-dilute regime, chain interactions become important, and the system starts showing
Eur. Phys. J. E (2012) 35: 104
tube-like dynamics as we have inferred from the rotational
diffusion in the previous section. In this model, a rod-like
particle cannot move as easily in the perpendicular direction compared with the parallel direction because of the
enhanced probability of encountering another particle. Accordingly a rod-like particle is observed to diffuse by rotating by an angle 1/Ar as it moves. Thus, D /D⊥ can
be expected to become of order Ar as density is increased
in the semi-dilute regime before the rods get caged. The
increasing D /D⊥ with Ar shown in fig. 10 are consistent
with this trend.
5 Conclusions
Using linked chains on a vibrated substrate, we have studied the diffusion of granular rod in two dimensions and find
significant effects due to their aspect ratio and density.
Multiple time scales in rotational as well the translational
diffusion degree of freedom are observed in the semi-dilute
regime. We observe that rolling of rods can have significant effect on their dynamics due to the low moment of
inertia of the rods about their long axis and frictional substrate. Due to presence of rolling, Dr was estimated to be
modified to a smaller value in semi-dilute regime. Further, the diffusion coefficient in perpendicular direction
was observed to be higher than the diffusion coefficient in
parallel direction in the dilute regime. In the semi-dilute
regime, chain interactions become important and the diffusion along the parallel direction is to be greater than the
perpendicular direction given by a factor which increases
with aspect ratio, consistent with tube-like dynamics. Finally, chains are observed to become caged at a lower area
fraction than spheres moving on a vibrated substrate.
Page 7 of 7
We thank Thomas Garnier for help with preliminary experiments. This work was supported by the National Science Foundation under NSF Grant No. CBET-0853943 and by the Jan
and Larry Landry Chair Fund.
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